Differentiating

Velocity is the rate of change of displacement, so if you had a function for displacement in terms of time you could differentiate it to find its velocity. Similarly velocity differentiates into acceleration.

Example

A train travels along a strait track between two stations. The train is at rest at both of these stations. At time $t$ hours, the trains distance from the station is given by $s = (90t^2 - 45t^3) \text{ km}$. Find the maximum speed of the train between the two stations.

Find the derivative of velocity

$$\begin{align} s &= 90t^2 - 45t^3\\ v &= \frac{ds}{dt} = 180t - 135t^2\\ \frac{dv}{dt} &= 180 - 270t \end{align}$$

Find the stationary point

$$\begin{align} 180 - 270t &= 0\\ -270t &= -180\\ t &= \frac{180}{270}\\ &= \frac{2}{3}\\ \end{align}$$

Find the velocity

$$\begin{align} v &= 180t - 135t^2\\ &= 180 \times \frac{2}{3} - 135 \times \left(\frac{2}{3}\right)^2\\ &= 60 \text{ kmh}^{-1} \end{align}$$